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Chapter 1
Aeroelastic analysis and design optimization
of cable-supported bridges
1.1 Introduction
Suspension and cable-stayed bridges are two important types of bridges frequently used
to connect great distances. Even though their origins and evolution processes are dif-
ferent, their analysis methods are very similar at present time. Thus the term, cable-
supported bridge is often used to refer to both types of structures and used as well in this
book in many occasions.
The first steel suspension bridge was built by James Finney,[1] who built one with
21-m span over the Jacobs Creek in 1801. Figure 1.1.1 shows the bridge designed by
Finney, which already included detailed structural elements used for a suspension bridge
today: the curved main cable, the vertical cables connected to the deck, the truss deck
and stone towers that were very common during almost all through the 19th century. It is
worth mentioning that Finney realized that the lateral spans did not need to be suspended
from the main cable, and as can be seen in the figure, only one of them is suspended from
the main cable.
More than 200 years have passed since then, and there is no new concept in the mod-
ern design of a suspension bridge as to the scheme in the figure. However, the span length
of important modern bridges is one hundred times longer than that of the first bridge,and as will be mentioned in the following chapter, there are many ambitious projects
underway.
The history of cable-stayed bridges is more recent than that of suspension bridges, and
it can be dated back to the beginning of the 20th century. Figures 1.1.2 and 1.1.3 show two
examples of this typology, in which we can observe the complex cable configuration: the
cables are not tied to the deck, but to the horizontal cable from which a group of vertical
cables connect to the deck. The reason for this uncommon solution is due to the fact that if
the cables connect to the deck directly, because of its inclination, it would generate large
compression on the deck, which would then cause buckling problem if it is not properlydesigned. They avoided the problem with this proposed solution, however, in the expense
of not only complicating the structure, but also the construction of the bridge, which
resulted in a very difficult job. Therefore this approach was finally abandoned. After the
World War II when the infrastructures of road communication needed to be reconstructed,
many cable-stayed bridges were built[2] such as the Strmsund Bridge erected in 1956
shown in Figure 1.1.4. Its structural configuration is a simple cable system that goes from
the towers to the deck making its calculation and construction easier.
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Figure 1.1.2: Cassagne Bridge (France).
Figure 1.1.3: Lazardrieux Bridge (France).
Figure 1.1.1: Sketch of suspension bridge by J. Finney.
Today both types of bridge have increased their capacity in such a dimension that the
suspension bridge over the Akashi Strait in Japan shown in Figure 1.1.5 achieved 1991
m of span length, and the cable-stayed Sutong Bridge over the Yanghse river in China in
Figure 1.1.6 reached 1088-m span length.
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For both types of bridges, even if they are small, the most important point that a
designer has to consider is not the service loads produced by automobile or railroadtraffic, but the seismic or wind loads, which are much more dangerous producing great
stresses in the material. Therefore, this book is dedicated to the study of cable-supported
bridges under wind actions, and especially to one of the most significant instability phe-
nomena that air flow can produce.
1.2 Aeroelastic phenomena
The air flow around a bridge not only causes the bridge to move but also changes the
loads that wind generates on the structure. There is an important interaction between the
wind pressure and the bridge deformation as an elastic body. These phenomena are called
aeroelastics, and the science that studies this interaction is called aeroelasticity[3].
Some of the most important phenomena, among all, taken place on cable-supported
bridge are as follows:
vortex shedding
flutter
buffeting cable vibration
Vortex shedding is produced when the air flow lines are modified for colliding with
a bridge deck. It takes place at certain wind velocities depending on the deck geometry
and produces vortices with alternating rotations, which then produce a vertical force. This
force changes in direction in each vortex, thereby causing vibrations of the deck. Since
such vibrations have limited amplitudes, they do not cause any catastrophes; however,
they might make pedestrians uncomfortable for walking or the velocity of circulating
traffic may have to be limited. They can also cause the material fatigue of the bridge.
Figure 1.1.4: Strmsund Bridge (Sweden).
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Figure 1.1.5: Akashi Bridge (Japan).
Flutter is a vibration produced on a bridge by an interaction with the wind flow, and
its amplitude is not only unlimited but also increases in each cycle. Once this aeroelastic
instability starts, it occasionally causes destruction of bridges. In the next chapter, we
will take a look at some examples of this phenomenon, which obviously could have been
avoided. The most famous example is that occurred to the bridge over the Tacoma Strait
in 1940 described in detail in the next chapter that triggered the development of aeroelas-
ticity in bridge engineering.
Figure 1.1.6: Sutong Bridge (China).
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Buffeting is a vibration phenomenon produced by a turbulent wind flow. Its effects on
a bridge are wind gusts of different average velocity. Although this phenomenon does
not cause instability, it should be limited just as in the case of vortex shedding so that the
service life of a bridge should be adequate or as long as possible.Cable vibration is a term for a group of fluidstructure interaction phenomena pro-
duced on the cables of cable-stayed bridges. They may occur in different velocity ranges
and influenced by rain. The deformation of cables can present very diverse trajectories
depending on those factors, their geometry, stiffness and mass. There has not been any
case of cable rupture by this phenomenon as of today; however, there have been events of
great amplitude and frequency vibrations so that corrector devices had to be installed. At
present time, there is no clear classification of a group of phenomena that causes this type
of vibrations, and it is still an undeveloped research area.
It should be clear that in spite of the importance of all the aeroelastic phenomena
described here, flutter is the most important since it may occasionally causes the total
destruction of a bridge. Therefore, it has been an objective of studies by different meth-
ods, and this book is dedicated exclusively to this aeroelastic instability.
1.3 Methodologies of flutter analysis
As mentioned briefly, studies on flutter of bridges started after the collapse of the Tacoma
Bridge in the 1940s. At that time, because of the non-existence of digital computers,the best way to acquire knowledge was by an experimental method. As same as for
aeronautical engineering, reduced bridge models were created and then tested in wind
tunnels under air flow at different velocities to observe the aeroelastic phenomena. In
order to obtain correct results, vibration frequencies used to the reduced model were
equally scaled as the geometry scale of the reduced model to the complete bridge[4]. This
method gave very fruitful results during various decades. There are very diverse ways
of performing a testing, and during the same experiment, one could obtain a wide range
of information on the structural responses of a reduced model. This will be described in
detail in this book. Figure 1.3.1 shows a test of the bridge over the Messina Strait usingone of those models.
The revolution of calculation tools produced by digital computers has been such a
magnitude that today computer programs are the most common calculation tools in engi-
neering. Therefore, it seems logical that the fluidstructure simulation in bridge engineer-
ing under the influence of aeroelastic phenomena is performed by computational fluid
mechanics[5](CFD), and wind-tunnel testing is not necessary any longer.
Aeronautic engineering had an earlier start in this regard in which CFD techniques
had opened up their way and became an important part of analysis and design tool for
engineers. They already have programs to resolve NavierStokes equations for wind flow
disturbed by an airfoil in a closed area to obtain essential pressure and velocity distribu-
tions as well as a set of aerodynamic parameters. The geometry of airfoil is designed all
for flight efficiency, which also helps the resolution of theproblem. Figure 1.3.2 shows
the aircraft A380, while Figure 1.3.3 shows the results of pressure field of a National
Advisory Committee for Aeronautics profile by this procedure.
However, the geometry of a bridge deck is affected by a wider range of factors than
that of an airplane. The most important task of a long-span bridge, at which most likely
aeroelastic phenomena take place, is let the vehicle traffic pass through it. Then a series
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Figure 1.3.1: Test of a reduced model of the Messina Bridge.
Figure 1.3.2: A380 aircraft.
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of security measures associated to the task must be added such as centre dividers between
two circulating directions or guide rails on the sides of the deck. In the case that pedestri-ans also use the bridge, there should be guide rails to separate and protect them from the
circulating traffic. All these additions lead to a poorer aerodynamic quality of the deck.
Figure 1.3.4 shows two bridge deck sections: one with truss section and the other with
box section. Even though the latter has a smoother section, the placement of guide rails
makes its aerodynamic behaviour less efficient. Therefore the fluidstructure interaction
of a bridge deck is more complex than an airfoil. A set of structural elements of truss sec-
tions or guide rails change wind flow lines in large quantity and different ways causing
turbulences of different sizes. Thus producing mathematical models of the phenomena
that take part is rather difficult, and the formulation of turbulence associated to each typeof bridge deck is still in an underdeveloped phase. This turbulence formulation is essen-
tial in order to adapt the NavierStroke equations appropriately. Although the number of
CFD applications in bridge engineering is yet very limited to resolve aeroelastic prob-
lems, for certain types of deck sections, excluding truss sections, they may be relatively
useful to determine the aerodynamic coefficients or to evaluate vortex shedding. How-
ever, the obtained results are not sufficiently accurate, and consequently we need to check
them experimentally, which reduces greatly the advantages of computational approach.
Nevertheless, this situation will be improved little by little in the future. Figure 1.3.5
shows a CFD simulation of vortex shedding at a cross section of a bridge deck.
Nonetheless, it seems obvious that the computer calculation technique should take
part in the study of general aeroelastic phenomena of cable-supported bridges, particu-
larly in the study of flutter. This results in a methodology that combines a wind-tunnel
testing with analysis by computer. This hybrid method tries to take advantage of both
experimental and numerical technologies as much as possible, and it will be described
in detail in later chapters. In the first experimental stage, the forces produced by wind
during the vibration of a bridge are obtained, and in the second computational stage,
the dynamic equilibrium of the bridge under such forces is studied. Finally a non-linear
Figure 1.3.3: Static pressure field on a NACA airfoil.
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eigenvalue problem is solved depending on the wind velocity. Since we can obtain the
flutter velocity of suspension and cable-stayed bridges precisely and efficiently in this
way, this method is well accepted for its great result quality and advantages over purely
experimental method. All the theories associated to this method are described in various
chapters, and examples of its applications on particular bridges, either under-construction
phase or already completed, are described in this book.
The advantages of this approach will be greater if, besides flutter velocity, it provides
deformation data of a bridge during the phenomenon, that is, if we can observe the geom-
etry changes during the vibration produced by wind flow. Although this method is not
designed for that purpose, the computational phase provides vibration frequencies and
modes of flutter velocity, which permits the use of this data combined with the actual
capacities of computer graphic presentation.
GENERAL CROSS-SECTION
(a)
(b)
CROSS-SECTION AT TOWERS
34 m.
3 m.
Figure 1.3.4: Types of bridge deck. (a) Truss bridge deck. (b) Box bridge deck.
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We can generate a digital model of an undeformed shape of a bridge as shown in Figure
1.3.6, in which we can see the accuracy of all the bridge elements. This type of represen-
tation is naturally more detailed than a reduced model for wind-tunnel testings.
With the digital model generated along with the knowledge of vibration frequency during
flutter and associated geometry of vibration modes, a computer animation can be gener-
ated to show vibration during this aeroelastic phenomenon[6].
In this way, the deformation of a bridge can be visualized clearly and graphically without
performing a wind-tunnel testing of a complete bridge. Figure 1.3.7 shows an image of
deformed bridge geometry during flutter by this procedure.
1.4 Sensitivity analysis: a design tool
The major part of the designing process in engineering for past decades was based on the
ability and good judgment of a designer, which progressively led him to refine an initial
prototype into a final design. This process was partially powered by digital computers;
for defining geometry of a structure such as a bridge or a dam, CAD programs might
have been utilized, and finite element programs might have been employed to check the
structural behaviour, which may have been supported by some testings as in the case of
wind for long-span bridges.
However, in each design phase when part of prototype did not function properly and
needed to be modified, the designer is the one who decided to make changes using some
simple rules gained by his experiences. The designer was free to decide the changes in
some aspects such as dimensions of some elements or their locations within the structure,
although some of the design properties may have been predetermined. For example, if
displacements of some areas of a bridge are too large, the inertia can be increased, but not
the span length because it is predetermined.
Figure 1.3.5: Velocity field of flow around a bridge deck showing vortex shedding.
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Figure 1.3.6: Digital model of the original Tacoma Bridge.
Figure 1.3.7: Deformation of the digital original Tacoma Bridge model under wind flow.
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In essence, a designer observes structural responses of a design in each phase and makes
changes in modifiable properties hoping that they are good enough.
This is, therefore, essentially a subjective process, and its efficiency all depends on the
designers capacity and the difficulty level of the problem. Consequently, the quality ofthe final result may vary greatly in each case.
There have been more rational tools around for a long time to assist engineers with these
tasks. They are based on so-called sensitivity analysis, which consists of obtaining the
derivatives of structural response with respect to modifiable structural properties to
improve the design. Let us take a look at a practical example of linear static structural
analysis to understand this approach.
Static equilibrium of a structure can be written as
Ku P
where Kis the stiffness matrix of a structure, uthe displacement vector and Pthe load
vector. All of them are functions of structural variables, for example, some mechanical
parameters
( ) ( ) ( )K x u x P x
where the vector x contains a group of modifiable structural properties to improve the
design, commonly called design variables.
By differentiating this expression,
d Ku P
or
1 1 1
n n n
i i i
i i ii i i
dx dx dxx x x
K u Pu K
= = =
+ =
For each variable, we have
i i ix x x
K u Pu K
or
i i ix x x
u P KK u
This expression is a system of equations that gives the derivatives of a displacementvector with respect to a design variable. By resolving it for each of them, we will get a
matrix as
1,...,
x xn
u u u
x
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which also can be written as
1
n
u
u
xu
x
x
Each element of this matrix contains two types of information. One is qualitative asso-
ciated with the sign of the derivative; if it is positive, in order to decrease the value of the
structural response, in this case, displacements of the structure, a designer has to decrease
the design variable value, while if the derivative is negative, he has to do the opposite.
Another is the quantitative value of the derivative, which gives an idea of how much
changes one can get from modifying the variable. If the absolute value of the derivative is
large, one can get significant changes by modifying that variable, while the modification
causes only minor changes if the derivative is small.
The first book on sensitivity analysis of structures was written by Haug et al.[7]Each
sensitivity analysis requires differentiation of the sate equation as in the case of the static
equilibrium equation described here. In this way, different types of structural problems
were studied, among which frequencies and natural vibration mode problems are found.
However, the sensitivity analysis associated to aeroelastic problems of bridge engineeringis only recent and it is a pioneer work of Jurado, Mosquera and Hernndez[8, 9]. We will
discuss the approach as well as various examples of its application to actual bridges of
great importance in several chapters of this book. This approach not only allows a bridge
designer to progress in the designing process faster and more rational, but also by observ-
ing carefully the results of sensitivity analysis, he can reflect on how much he could have
anticipated the results by his intuition and experience. This makes an engineer mature and
to improve his design skills.
1.5 Optimum design in engineering: application to bridge
aeroelasticity
The techniques of design optimization in engineering have been around for the last dec-
ades. The first work of L. Schmit was published in 1960[10], in which he utilized skilfully
a formulation of this method. It consisted of combining the power of finite element struc-
tural analysis already developed at that time with optimization algorithms, some of which
already existed. The objective of optimization techniques was to perform design improve-
ment process automatically, which was up till then performed subjectively by designers.
An iterative process was used just as in the conventional process, but rather in a system-
atic manner. The process started with an initial design whose quality was checked to see
if all the requirements were satisfied. When some of the requirements were not satisfied,
instead of modifying the design arbitrarily, the process kept giving data of the actual
design to an optimization algorithm that produced a new design, which was improved in
quality from the previous one. By reiterating the process, the solution finally converged
to a design that satisfied all the required constraints in the best way. Figure 1.5.1 shows
schematically this method that will be described in detail in the corresponding chapter.
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After the slow acceptance described magnificently by the author[11], little by little the
method was gaining enthusiastic supports and acceptations in the academic world as well
as in industries. Since then there have been numerous texts that describe the fundamental
of these techniques and some of them describe practical applications in wide range of
fields[1216]. Today, there are commercial codes with optimization modules utilized con-
tinuously by automobile or aerospace companies, while in other sectors of industries, this
technology is not employed very much.The use of this design method not only provides better quality of the final design that
would give greater competitiveness to the company but also helps a designer with his
task. When the designing process is finished, a designer should think about the differ-
ences between the initial design and that finally produced by the optimization method.
From this reflection, he should observe the changes produced and think which ones he
was able to anticipate and which ones not. This will improve his skills as a designer, and
we can see the optimization technique as a design course that a designer can take while
he is performing his job.
In summary, we are not trying to substitute a designer with these optimization tech-niques, which would be impossible because of the complexity of real problems, but rather
intending to help a designer not to fall into false steps that can be very probable for a
design with great complexity.
Although it is appropriate to apply this technique to aeroelastic problems in bridge
engineering, it is only recent to utilize them for optimization of cable-supported
bridges including constraints associated with fluidstructure interaction. Specifically,
Nieto, Hernndez and Jurado were pioneers in dealing with optimum design of sus-
pension bridges considering flutter phenomenon in the problem. The last two chapters
of this book describe the formulation of this problem as well as its application to a
real bridge.
Because of the complexity of obtaining flutter velocity, which requires successive itera-
tions for being a non-linear problem as well as the iterative characteristics of the opti-
mization itself, the application of this technique to aeroelastic problems demands great
calculation capacity. Nevertheless, this issue will surely be less and less important with
the fast evolution of computers since calculations can be carried out by sufficiently pow-
erful machines or computer clusters that can run in parallel for distributing calculation
loads.
Figure 1.5.1: Flowchart of design optimization technique.
INITIAL DESIGNSTRUCTURAL
ANALYSIS
YES
FINAL DESIGN
NO
OPTIMIZATION
PROCEDURE
OPTIMUM
DESIGN
MODIFIED
DESIGN
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1.6 References
[1] Brown D. J. [1993]Bridges, Macmillan, New York.
[2] Gimsing N. J. [1997] Cable Supported Bridge, Concept and Design, John Wiley &Sons, New York.
[3] Simiu E., Scanlan R. H. [1996] Wind Effects on Structures, John Wiley & Sons,
New York.
[4] Dyrbye C., Hansen S. O. [1997] Wind Loads on Structures, John Wiley & Sons,
New York.
[5] Wendt J. F. (ed.) [2009] Computational Fluid Dynamics, Springer, Berlin.
[6] Nieto F., Hernndez S., Jurado J. A. [2009] Virtual wind tunnel. An alternative
approach for the analysis of bridge behaviour under wind loads. International
Journal of Advances in Engineering Software, Vol. 40, No. 3, pp 229235. [7] Haug E. J., Choi K. K., Komkov V. [1986] Design sensitivity analysis of struc-
tural systems, Mathematics in Science and Engineering, Vol. 177, Academic Press,
New York.
[8] Jurado J. A., Hernndez S. [2004] Sensitivity analysis of bridge flutter with respect
to mechanical parameters of the deck. Structural and Multidisciplinary Optimiza-
tion, Vol. 27, No. 4, pp. 272283.
[9] Mosquera A., Hernndez S, Jurado J. A. [2003] Analytical sensitivity analysis of
aeroelastic performance of suspension bridges under construction. 11th Interna-
tional Conference on Wind Engineering. Lubbock, TX, USA.[10] Schmit L. A. [1960] Structural design by systematic synthesis. Second Conference
on Electronic Computation. ASCE, Pittsburg, PA, pp. 105132, 1960.
[11] Schmit, L. A. [1981] Structural synthesis. Its genesis and developement. AIAA
Journal, Vol. 19, No. 10, pp. 12491263.
[12] Vanderplaats, G. N. [2001] Numerical Optimization Techniques for Engineering
Design. VR&D, Colorado Springs.
[13] Haftka R.T., Gurdal Z. [1992] Elements of Structural Optimization. Kluwer,
Dordrecht.
[14] Arora J. S. [2005]Introduction to Optimum Design. Elsevier, The Netherlands.[15] Belegundu A. D., Chandrupatla T. R [1999] Optimization Concepts and Applica-
tions in Engineering. Prentice-Hall, New York.
[16] Hernndez S., Fontan A. N. [2002]Practical Applications of Design Optimization.
WIT Press, Southampton.